Limit cycles and chaos in the hybrid atom-optomechanics system

We consider atoms in two different periodic potentials induced by different lasers, one of which is coupled to a mechanical membrane via radiation pressure force. The atoms are intrinsically two-level systems that can absorb or emit photons, but the dynamics of their position and momentum are treated classically. On the other hand, the membrane, the cavity field, and the intrinsic two-level atoms are treated quantum mechanically. We show that the mean excitation of the three systems can be stable, periodically oscillating, or in a chaotic state depending on the strength of the coupling between them. We define regular, limit cycle, and chaotic phases, and present a phase diagram where the three phases can be achieved by manipulating the field-membrane and field-atom coupling strengths. We also computed other observable quantities that can reflect the system’s phase such as position, momentum, and correlation functions. Our proposal offers a new way to generate and tune the limit cycle and chaotic phases in a well-established atom-optomechanics system.

www.nature.com/scientificreports/ In this paper, we consider the atoms trapped in a double well-like potential created by two lasers, where the wavelength of one is half of the other. One of the lasers is filtered and enters a cavity where it couples to a membrane, which in turn affects its optical path. This way, the atoms will have both time-dependent and fixed potentials. In this configuration, relevant interactions include optomechanical coupling between the field and the membrane as well as light-matter coupling between the field and the atoms. The position and momentum of the atoms are treated classically, with their dynamical equations coupled to a quantum master equation characterizing the membrane, the cavity field, and the intrinsic degrees of freedom of the atoms (two-level systems). We show that by tuning the strength of the optomechanical and light-matter coupling, the system can be in regular, limit cycle, or chaotic phases. We also computed experimentally familiar quantities such as the first and second-order correlation functions in different phases.

Model
Consider two-level atoms moving in an adjusted gauge field optical lattice, which is coupled to a membrane through the coherently driven cavity field as shown in Fig. 1. The cold atoms are trapped by two lasers with different wavelengths, giving two optical lattice potentials with different periods. The Hamiltonian ( = 1 ) describing the membrane, cavity field, and two-level atom, in a frame rotating with the driving frequency ω l and with rotating-wave approximation, is written as where ω m is the frequency of the membrane, � c = ω l − ω c and � a = ω l − ω a are the detuning for the cavity and atom, respectively. ω c denotes the cavity frequency, ω a the atomic transition frequency and ω l the frequency of the laser driving the cavity with strength η . The atom couples to the cavity (Jaynes-Cummings type) with strength g ac , while the optomechanical coupling 18 between the membrane and cavity is denoted by g mc . The optical lattice has a mode function sin(2x) , where x is the atomic position, which is in units of the inverse cavity wave number. The annihilation operators for the cavity, atom, and membrane are denoted by â , σ − , and b , respectively.
The decays in the system are modelled by the Liouvillians and can be considered as Lindblad terms where γ µ is the dissipation rate of the membrane ( µ = m ), cavity field (c), and two-level atom (a). Note that Ô denotes the corresponding annihilation operator of each system. As the initial state, we use uncorrelated states of the form ρ = ρ m ⊗ ρ c ⊗ ρ a , where ρ m , ρ c , and ρ a represent the density matrix for the membrane, cavity field, and atom. The evolution follows the quantum master equation: In addition, we have classical differential equations of the atomic motion obtained from the Ehrenfest theorem: ẋ = ∂�H�/∂p and ṗ = −∂�H�/∂x , where the observables x and p are treated simply as numbers in the classical regime. For this classical motion, the atom is situated in two potentials such that its Hamiltonian reads where m is the mass of the atom, V 0 is the depth of the optical lattice, and V 1 sin(x) is the external periodic potential. We stress that only one of the potentials ( V 0 ) ends up being coupled to the membrane. The equations of motion for the atom, taking into account the Hamiltonians H and H a , are written as  www.nature.com/scientificreports/ where ω r = 1/(2m) is the recoil frequency. As initial conditions, we take x(0) = −1 and p(0) = 0 . We note that the quantum observable ℜ �â †σ − � updates the classical dynamics, while the latter affects the quantum dynamics via the change in the optical path, and hence, the mode function sin(2x) . We show below with suitable parameters, that this quantum-classical coupled dynamics can produce regular, limit cycle, and chaotic phases. See also the Appendix for calculations of the dynamics using the quantum trajectory method.

Different dynamical behaviors
The hybrid system has quantum and classical parts that are treated differently but are coupled to each other. For the quantum part, all three systems are also coupled, and consequently, we note that a particular phase in one system is an indication of the same phase in others. In what follows, we define three phases based on the dynamical behavior of mean excitations (either of the membrane n m , cavity field n c , or two-level atom �σ +σ − � ): 1. Regular phase. The excitation of each system will have a stable value after a long evolution time, as shown in Fig. 2a1-a3. In this case, the motion of the atom and the membrane will converge to a point in the phasespace diagram, as shown in Fig. 3a1,b1. 2. Limit cycle phase. The excitation of each system will oscillate periodically around a certain value, as shown in Fig. 2b1-b3. Here, after a certain time, the motion of the atom and the membrane will continue to orbit a point in the phase-space diagram, see Fig. 3a2,b2. 3. Chaotic phase. The excitation of each system will show random oscillation, as shown in Fig. 2c1-c3. The motion of the atom and the membrane in the phase-space diagram will exhibit random orbits around two attractors, see Fig. 3a3,b3.
Remarkably, the quantum-classical coupled dynamics can produce all three phases, by simply tuning the strength of the optomechanical ( g mc ) and light-matter ( g ac ) coupling, see Fig. 2.
Furthermore, the atomic motion (x, p) and expectation value of quadratures, e.g., for the membrane ( Fig. 3. For the atom, as the period of one potential is twice the other (see Eq. (4)), the depths V 0 and V 1 allow for a double well-like Figure 2. The mean excitation of the membrane mode (first column), the cavity mode (second column) and the atom (third column). The parameters used are γ c /γ a = 0.5, γ m /γ a = 2, η/γ a = 5, V 0 /γ a = 20, V 1 /γ a = 40, ω r /γ a = 1, � c /γ a = − 1, and � a /γ a = − 2. Specific coupling parameters for different phases are given by (a1)-(a3) g ac /γ a = 0.5, g mc /γ a = 2; (b1)-(b3) g ac /γ a = 2, g mc /γ a = 2.5; and (c1)-(c3) g ac /γ a = 2, g mc /γ a = 2. www.nature.com/scientificreports/ potential shape, which consequently gives three optimum points, two of which are stable. The position of the two stable points are symmetric with respect to, e.g., x = −π/2 where the stronger potential V 1 sin(x) has the lowest energy. Thus, the steady momentum is always zero in the regular phase, whereas it is oscillating around zero in other phases. The atomic motion will converge to one of the stable points in the regular phase while the trajectory will form a closed circle in the limit cycle phase. When the system is in the chaotic phase, there are two attractors in the phase-space diagram and the motion of the atom is unpredictable. At the same time, the behaviors of the quantum degrees of freedom reflect that of the classical ones (x, p) of the atom, see the second column of Fig. 3 for the membrane's quadratures. The motion of the membrane will have non-zero momentum in the steady-state regime. Below we shall introduce quantities to indicate the phase of the system, and finally, obtain a phase transition diagram for varying values of the optomechanical and light-matter coupling strengths. We also computed the first and second order correlation functions G (1) (τ ) and G (2) (τ ) that are standardly measured in experiments. See the Appendix for details. As expected, the behaviors of these correlation functions follow that of the mean excitation in the corresponding phases.

Quantification and classification of the phases
Here we shall present a way to numerically classify the phases previously described. In particular, we used two quantities, where one is recognizing the regular phase and the other the chaotic phase. Consequently, this method classifies all three possible phases in the phase transition diagram, which we will present below.
The regular phase transition. Recognizing the regular phase is straightforward as the mean excitation of all the systems will go towards a constant value, see Fig. 2. Here, after a long evolution time, one can choose a time range and compute R = max (�n µ �) − min (�n µ �) . The regular phase is given for R < ǫ , where ǫ is a small constant.
The 0-1 test for the chaotic phase transition. The system in the chaotic phase will have a very different dynamical behavior, which can be tested by the regression or correlation method 54 . Here, relevant functions are defined such that we can apply the above tests to our system. First, we take new translation components ( x ac , p ac ) and θ c as follows Figure 3. The motion of the atom (first row) and the membrane (second row) in the phase-space diagram. The parameters are given by γ c /γ a = 0.5, γ m /γ a = 2, η/γ a = 5, V 0 /γ a = 20, V 1 /γ a = 40, ω r /γ a = 1, � c /γ a = − 1, and � a /γ a = − 2. For panels (a1,b1): g ac /γ a = 0.5, g mc /γ a = 2; (a2,b2): g ac /γ a = 2, g mc /γ a = 2.5; and (a3,b3): g ac /γ a = 2, g mc /γ a = 2. www.nature.com/scientificreports/ where n = 1, 2, . . . , N denotes the time index, φ(n) is a dynamical quantity, here taken as x(n) + p(n) , and ν is a fixed constant [0, π] . The initial state of p ac , x ac and θ c are zero and they are updated by the position and the momentum of atoms. The quantities q ac and p ac are bounded if the dynamical behavior is regular, while in the chaotic phase they will behave asymptotically. The translation components resulting from the hybrid atomoptomechanical system are shown in Fig. 4. The regular and limit cycle phases have bounded states for ( x ac , p ac ) as shown in Fig. 4a,b. However, they become unbounded in the chaotic phase, see Fig. 4c,d, showing the pattern of fractals. Given dynamical components ( x ac , p ac ), the mean square displacement is defined as where n ≪ N is required. The test for chaos is based on the growth rate of M c (n) as a function of n. A modified mean square displacement D c (n) that exhibits the same asymptotic growth as M c (n) , but with better convergence properties is given by where the oscillation term V osc is defined as V osc = (E φ ) 2 (1 − cos(nν))/(1 − cos(ν)), and the expectation E φ is given by E φ = lim N→∞ 1 N N j=1 φ(j) . Note that the cut-off index n cut needs to be large enough such that the error of D c (n cut ) is close to zero.
The phase transition diagram. The three phases are characterized by the two tests described above (regular and chaotic phase transition tests). The phase diagram for the two tests are plotted separately, see the Appendix. Here, we combine the diagrams, see Fig. 5, which shows the three phases for different coupling strengths. When the atom-cavity coupling is close to zero, only the regular phase exists with the balance of the rates of the decay and the drive. With the increase of g ac , the limit cycle phase will appear with periodic evolution of the interaction strength g ac ℜ �â †σ − � . For further increase of g ac the system reaches the chaotic phase. Remarkably, the coupling between the cavity and the membrane g mc also plays an important role in the limit cycle and chaotic phases. If g mc is too small compared to g ac the model can be simplified to an atom cooling model and the membrane's oscillations can be ignored. On the contrary, if g ac is too small, the system can be transformed to an optomechanical model and the atoms can be ignored. The competition of the coupling strengths allows the system to have a rich phase diagram. The regular-limit cycle transitions start to happen at g mc /γ a ∼ 0.8 and g ac /γ a ∼ 1 , see Fig. 5. In other words, exemplary parameters for obtaining the regular phase are g mc /γ a < 0.8 and g ac /γ a < 1 . For observing the other two phases, exemplary parameters are given as follows. For the limit cycle: g mc /γ a = 3 and g ac /γ a = 2 , while for the chaotic phase: g mc /γ a = 1 and g ac /γ a = 3.

Conclusion
We theoretically considered a hybrid atom-optomechanics system to realize different dynamical phases by exploiting the competition of the coupling strength of the cavity and atoms, and that of the cavity and membrane. The atoms experience two potentials, including one that may be static, periodically oscillating, or randomly oscillating. The coupling of the cavity mode and the membrane allows them to have similar behavior, where the whole system can exhibit a regular, limit cycle, or chaotic phase. These three phases are distinguished after evolving quantities from the system for a sufficiently long time, where we performed regular and chaotic phase transition tests. Our study motivates possibilities such as indirect measurements or inference of the states of the membrane from the states of the atoms, or vice versa. It is also interesting to study the dynamics of quantum correlations (e.g., entanglement) between different subsystems in different phases for potential quantum information applications such as quantum state transfer and metrology similar to what has been achieved in optomechanics 18 .